Chapter 2. Hypothesis testing in one population


 Debra Henry
 7 years ago
 Views:
Transcription
1 Chapter 2. Hypothesis testing in one population Contents Introduction, the null and alternative hypotheses Hypothesis testing process Type I and Type II errors, power Test statistic, level of significance and rejection/acceptance regions in upper, lower and twotail tests Test of hypothesis: procedure pvalue Twotail tests and confidence intervals Examples with various parameters Power and sample size calculations
2 Chapter 2. Hypothesis testing in one population Learning goals At the end of this chapter you should be able to: Perform a test of hypothesis in a onepopulation setting Formulate the null and alternative hypotheses Understand Type I and Type II errors, define the significance level, define the power Choose a suitable test statistic and identify the corresponding rejection region in upper, lower and twotail tests Use the pvalue to perform a test Know the connection between a twotail test and a confidence interval Calculate the power of a test and identify a sample size needed to achieve a desired power
3 Chapter 2. Hypothesis testing in one population References Newbold, P. Statistics for Business and Economics Chapter 9 ( ) Ross, S. Introduction to Statistics Chapter 9
4 Test of hypothesis: introduction A test of hypothesis is a procedure that: is based on a data sample and allows us to make a decision about a validity of some conjecture or hypothesis about the population X, typically the value of a population parameter θ (θ can be any of the parameters we covered so far: µ, p, σ 2, etc) This hypothesis, called a null hypothesis (H 0 ): Can be thought of as a hypothesis being supported (before the test is carried out) Will be believed unless sufficient contrary sample evidence is produced When sample information is collected, this hypothesis is put in jeopardy, or tested
5 The null hypothesis: examples 1. A manufacturer who produces boxes of cereal claims that, on average, their contents weigh at least 20 ounces. To check this claim, the contents of a random sample of boxes are weighed and inference is made. Population: X = weight of a box of cereal (in oz) µ 0 z} { Null hypothesis, H 0 : µ 20 'SRS Does sample data produce evidence against H 0? 2. A company receiving a large shipment of parts accepts their delivery only if no more than 50% of the parts are defective. The decision is based on a check of a random sample of these parts. Population: X = 1 if a part is defective and 0 otherwise X Bernoulli(p), p = proportion of defective parts in the entire shipment p 0 z} { Null hypothesis, H 0 : p 0.5'SRS Does sample data produce evidence against H 0?
6 Null hypothesis, H 0 States the assumption to be tested We begin with the assumption that the null hypothesis is true (similar to the notion of innocent until proven guilty) Refers to the status quo Always contains a =, or sign (closed set) May or may not be rejected Simple hypothesis (specifies a single value): H 0 : µ = µ 0 z} { 5, H 0 : p = p 0 z} { 0.6, H 0 : σ 2 = Parameter space under this null: Θ 0 = {θ 0} Composite hypothesis (specifies a range of values): H 0 : µ σ 2 0 z} { 9 In general: H 0 : θ = θ 0 µ 0 p 0 z} { z} { 5, H 0 : p 0.6 In general: H 0 : θ θ 0 or H 0 : θ θ 0 Parameter space under this null: Θ 0 = (, θ 0] or Θ 0 = [θ 0, )
7 Alternative hypothesis, H 1 If the null hypothesis is not true, then some alternative must be true, and in carrying out a hypothesis test, the investigator formulates an alternative hypothesis against which the null hypothesis is tested. The alternative hypothesis H 1: Is the opposite of the null hypothesis Challenges the status quo Never contains =, or sign May or may not be supported Is generally the hypothesis that the researcher is trying to support Onesided hypothesis: (uppertail) H 1 : µ > 5 (lowertail) H 0 : p < 0.6 In general: H 1 : θ > θ 0 or H 1 : θ < θ 0 Parameter space under this alternative: Θ 1 = (θ 0, ) or Θ 1 = (, θ 0) Twosided hypothesis (twotail): H 1 : σ 2 9 In general: H 1 : θ θ 0 Parameter space under this alternative: Θ 1 = (, θ 0) (θ 0, )
8 The alternative hypothesis: examples 1. A manufacturer who produces boxes of cereal claims that, on average, their contents weigh at least 20 ounces. To check this claim, the contents of a random sample of boxes are weighed and inference is made. Population: X = weight of a box of cereal (in oz) Null hypothesis, H 0 : µ 20 versus Alternative hypothesis, H 1 : µ < 20'SRS Does sample data produce evidence against H 0 in favour of H 1? 2. A company receiving a large shipment of parts accepts their delivery only if no more than 50% of the parts are defective. The decision is based on a check of a random sample of these parts. Population: X = 1 if a part is defective and 0 otherwise X Bernoulli(p), p = proportion of defective parts in the entire shipment Null hypothesis, H 0 : p 0.5 versus Alternative hypothesis, H 1 : p > 0.5'SRS Does sample data produce evidence against H 0 in favour of H 1?
9 Hypothesis testing process xyyxxxxyy Population: X = height of a UC3M student (in m) Claim: On average, students are shorter than 1.6 Hypotheses: H 0 : µ 1.6 versus H 1 : µ > 1.6 'SRS yyxx Sample: Suppose the sample mean height is 1.65 m, x = 1.65 Is it likely to observe a sample mean x = 1.65 if the population mean is µ 1.6? If not likely, reject the null hypothesis in favour of the alternative.
10 Hypothesis testing process Having specified the null and alternative hypotheses and collected the sample information, a decision concerning the null hypothesis (reject or fail to reject H 0 ) must be made. The decision rule is based on the value of a distance between the sample data we have collected and those values that would have a nigh probabiilty under the null hypothesis. This distance is calculated as the value of a socalled test statistic (closely related to the pivotal quantities we talked about in Chapter 1). We will discuss specific cases later on. However, whatever decision is made, there is some chance of reaching an erroneous conclusion about the population parameter, because all that we have available is a sample and thus we cannot know for sure if the null hypothesis is true or not. There are two possible states of nature and thus two errors can be committed: Type I and Type II errors.
11 Type I and Type II errors, power Type I Error: to reject a true null hypothesis. A Type I error is considered a serious type of error. The probability of a Type I Error is equal to α and is called the significance level. α = P(reject the null H 0 is true) Type II Error: to fail to reject a false null hypothesis. The probability of a Type II Error is β. β = P(fail to reject the null H 1 is true) power: is the probability of rejecting a null hypothesis (that is false). power = 1 β = P(reject the null H 1 is true) Actual situation Decision H 0 true H 0 false Do not No error Type II Error Reject H 0 (1 α) (β) Reject Type I error No Error H 0 (α) (1 β = power)
12 Type I and Type II errors, power Type I and Type II errors can not happen at the same time Type I error can only occur if H0 is true Type II error can only occur if H0 is false If the Type I error probability (α), then the Type II error probability β All else being equal: β when the difference between the hypothesized parameter value and its true value β when α β when σ β when n The power of the test increases as the sample size increases For θ Θ1 power(θ) = 1 β For θ Θ0 power(θ) α
13 Test statistic, level of significance and rejection region Test statistic, T Allows us to decide if the sample data is likely or unlikely to occur, assuming the null hypothesis is true. It is the pivotal quantity from Chapter 1 calculated under the null hypothesis. The decision in the test of hypothesis is based on the observed value of the test statistic, t. The idea is that, if the data provide an evidence against the null hypothesis, the observed test statistic should be extreme, that is, very unusual. It should be typical otherwise. In distinguishing between extreme and typical we use: the sampling distribution of the test statistic the significance level α to define socalled rejection (or critical) region and the acceptance region.
14 Test statistic, level of significance and rejection region Rejection region (RR) and acceptance region (AR) in size α tests: Uppertail test H 1 : θ > θ 0 α RR α = {t : t > T α} AR α = {t : t T α} AR CRITICAL VALUE RR Lowertail test H 1 : θ < θ 0 α RR α = {t : t < T 1 α} AR α = {t : t T 1 α} RR CRITICAL VALUE AR Twotail test H 1 : θ θ 0 RR α = {t : t < T 1 α/2 or t > T α/2 } AR α = {t : T 1 α/2 t T α/2 } α 2 α 2 RRCRITICAL AR CRITICALRR VALUE VALUE
15 Test statistics Let X n be a s.r.s. from a population X with mean µ and variance σ 2, α a significance level, z α the upper α quantile of N(0,1), µ 0 the population mean under H 0, etc. Parameter Assumptions Test statistic RRα in twotail test Mean Variance Normal data Known variance Nonnormal data Large sample Bernoulli data Large sample Normal data Unknown variance Normal data X µ 0 σ/ N(0, 1) n X µ 0 ˆσ/ ap. N(0, 1) n ˆp p 0 p p0 (1 p 0 )/n ap. N(0, 1) jz : X µ 0 s/ n t n 1 (n 1)s 2 σ 2 0 χ 2 n 1 >< χ 2 : 8 z 9 z } { >< x µ 0 z : σ/ < z 1 α/2 or x µ >= 0 n σ/ n > z α/2 >: >; j x µ z : 0 ˆσ/ n < z 1 α/2 or x µ ff 0 ˆσ/ n > z α/2 ff ˆp p p 0 p0 (1 p 0 )/n < z 1 α/2 or ˆp p p 0 p0 (1 p 0 )/n > z α/2 8 t 9 z } { >< x µ 0 t : s/ < t n 1;1 α/2 or x µ >= 0 n s/ n > t n 1;α/2 >: >; 8 9 χ 2 z } { (n 1)s 2 σ 2 0 < χ 2 (n 1)s2 or n 1;1 α/2 σ 0 2 > χ 2 n 1;α/2 >= St. dev. Normal data (n 1)s 2 σ 2 0 χ 2 n 1 >: ( χ 2 : (n 1)s 2 σ 2 0 >; ) < χ 2 (n 1)s2 or n 1;1 α/2 σ 0 2 > χ 2 n 1;α/2 Question: How would you define RR α in upper and lowertail tests?
16 Test of hypothesis: procedure 1. State the null and alternative hypotheses. 2. Calculate the observed value of the test statistic (see the formula sheet). 3. For a given significance level α define the rejection region (RR α ). Reject H0, the null hypothesis, if the test statistic is in RR α and fail to reject H 0 otherwise. 4. Write down the conclusions in a sentence.
17 Uppertail test for the mean, variance known: example Example: 9.1 (Newbold) When a process producing ball bearings is operating correctly, the weights of the ball bearings have a normal distribution with mean 5 ounces and standard deviation 0.1 ounces. The process has been adjusted and the plant manager suspects that this has raised the mean weight of the ball bearings, while leaving the standard deviation unchanged. A random sample of sixteen bearings is selected and their mean weight is found to be ounces. Is the manager right? Carry out a suitable test at a 5% level of significance. Population: X = weight of a ball bearing (in oz) X N(µ, σ 2 = ) Test statistic: Z = X µ 0 σ/ N(0, 1) n Observed test statistic: 'SRS: n = 16 Sample: x = Objective: test µ 0 z} { H 0 : µ = 5 against H 1 : µ > 5 (Uppertail test) σ = 0.1 µ 0 = 5 n = 16 x = z = x µ0 σ/ n = / 16 = 1.52
18 Uppertail test for the mean, variance known: example Example: 9.1 (cont.) Rejection (or critical) region: RR 0.05 = {z : z > z 0.05 } = {z : z > 1.645} z= 1.52 Since z = 1.52 / RR 0.05 we fail to reject H 0 at a 5% significance level. N(0,1) density AR z α = Conclusion: The sample data did not provide sufficient evidence to reject the claim that the average weight of the bearings is 5oz. RR
19 Definition of pvalue It is the probability of obtaining a test statistic at least as extreme ( or ) as the observed one (given H 0 is true) Also called the observed level of significance It is the smallest value of α for which H 0 can be rejected Can be used in step 3) of the testing procedure with the following rule: If pvalue < α, reject H0 If pvalue α, fail to reject H0 Roughly: small pvalue  evidence against H0 large pvalue  evidence in favour of H0
20 pvalue pvalue when t is the observed value of the test statistic T : Uppertail test H 1 : θ > θ 0 test stat p value =area pvalue = P(T t) Lowertail test H 1 : θ < θ 0 pvalue = P(T t) Twotail test H 1 : θ θ 0 pvalue = P(T t ) + P(T t ) p value =area test stat test stat p value =left+right areas test stat
21 pvalue: example Example: 9.1 (cont.) Population: X = weight of a ball bearing (in oz) X N(µ, σ 2 = ) 'SRS: n = 16 Sample: x = Objective: test µ 0 z} { H 0 : µ = 5 against H 1 : µ > 5 (Uppertail test) Test statistic: Z = X µ 0 σ/ N(0, 1) n Observed test statistic: z = 1.52 N(0,1) density pvalue = P(Z z) = P(Z 1.52) = where Z N(0, 1) Since it holds that pvalue = α = 0.05 we fail to reject H 0 (but would reject at any α greater than , e.g., α = 0.1). z= 1.52 p value =area
22 The pvalue and the probability of the null hypothesis 1 The pvalue: is not the probability of H0 nor the Type I error α; but it can be used as a test statistic to be compared with α (i.e. reject H 0 if pvalue < α). We are interested in answering: How probable is the null given the data? Remember that we defined the pvalue as the probability of the data (or values even more extreme) given the null. We cannot answer exactly. But under fairly general conditions and assuming that if we had no observations Pr(H 0) = Pr(H 1) = 1/2, then for pvalues, p, such that p < 0.36: ep ln(p) Pr(H 0 Observed Data) 1 ep ln(p). 1 Selke, Bayarri and Berger, The American Statistician, 2001
23 The pvalue and the probability of the null hypothesis This table helps to calibrate a desired pvalue as a function of the probability of the null hypothesis: pvalue Pr(H 0 Observed Data) For a pvalue equal to 0.05 the null has a probability of at least 29% of being true While if we want the probability of the null being true to be at most 5%, the pvalue should be no larger than
24 Confidence intervals and twotail tests: duality A twotail test of hypothesis at a significance level α can be carried out using a (twotail) 100(1 α)% confidence interval in the following way: 1. State the null and twosided alternative H 0 : θ = θ 0 against H 1 : θ θ 0 2. Find a 100(1 α)% confidence interval for θ 3. If θ 0 doesn t belong to this interval, reject the null. If θ 0 belongs to this interval, fail to reject the null. 4. Write down the conclusions in a sentence.
25 Twotail test for the mean, variance known: example Example: 9.2 (Newbold) A drill is used to make holes in sheet metal. When the drill is functioning properly, the diameters of these holes have a normal distribution with mean 2 in and a standard deviation of 0.06 in. To check that the drill is functioning properly, the diameters of a random sample of nine holes are measured. Their mean diameter was 1.95 in. Perform a twotailed test at a 5% significance level using a CIapproach. Population: 100(1 α)% = 95% confidence X = diameter of a hole (in inches) interval for µ: X N(µ, σ 2 = ) ( x 1.96 n σ ) 'SRS: n = 9 Sample: x = 1.95 Objective: test µ 0 {}}{ H 0 : µ = 2 against H 1 : µ 2 (Twotail test) CI 0.95 (µ) = = ( ) 9 = (1.9108, ) Since µ 0 = 2 / CI 0.95 (µ) we reject H 0 at a 5% significance level.
26 Twotail test for the proportion: example Example: 9.6 (Newbold) In a random sample of 199 audit partners in U.S. accounting firms, 104 partners indicated some measure of agreement with the statement: Cash flow from operations is a valid measure of profitability. Test at the 10% level against a twosided alternative the null hypothesis that onehalf of the members of this population would agree with the preceding statement. Population: X = 1 if a member agrees with the Test statistic: statement and 0 otherwise Z = ˆp p 0 X Bernoulli(p) approx. N(0, 1) p0(1 p 0)/n Observed test statistic: 'SRS: n = 199 large n Sample: ˆp = = Objective: test p 0 {}}{ H 0 : p = 0.5 against H 1 : p 0.5 (Twotail test) p 0 = 0.5 n = 199 ˆp = z = ˆp p 0 p0 (1 p 0 )/n = (1 0.5)/199 = 0.65
27 Twotail test for the proportion: example Example: 9.6 (cont.) Rejection (or critical) region: RR 0.10 = {z : z > z 0.05 } {z : z < z 0.05 } = {z : z > 1.645} {z : z < 1.645} z= 0.65 Since z = 0.65 / RR 0.10 we fail to reject H 0 at a 10% significance level. N(0,1) density RR z α 2 = 1.645AR zα = RR 2 Conclusion: The sample data does not contain sufficiently strong evidence against the hypothesis that onehalf of all audit partners agree that cash flow from operations is a valid measure of profitability.
28 Lowertail test for the mean, variance unknown: example Example: 9.4 (Newbold, modified) A retail chain knows that, on average, sales in its stores are 20% higher in December than in November. For a random sample of six stores the percentages of sales increases were found to be: 19.2, 18.4, 19.8, 20.2, 20.4, Assuming a normal population, test at a 10% significance level the null hypothesis (use a pvalue approach) that the true mean percentage sales increase is at least 20, against a onesided alternative. Population: X = stores increase in sales from Nov to Dec (in %s) X N(µ, σ 2 ) σ 2 unknown 'SRS: n = 6 small n Sample: x = = 19.5 s 2 = (19.5)2 6 1 = Objective: test µ 0 z} { H 0 : µ 20 against H 1 : µ < 20 (Lowertail test) Test statistic: T = X µ 0 s/ n tn 1 Observed test statistic: µ 0 = 20 n = 6 x = 1.95 s = = t = x µ0 s/ n = / 6 = 1.597
29 Lowertail test for the mean, variance unknown: example Example: 9.4 (cont.) pvalue = P(T 1.597) (0.05, 0.1) because t 5;0.05 t 5;0.10 { }} { { }} { < < Hence, given that pvalue < α = 0.1 we reject the null hypothesis at this level. p value =area t= t n 1 density Conclusion: The sample data gave enough evidence to reject the claim that the average increase in sales was at least 20%. pvalue interpretation: if the null hypothesis were true, the probability of obtaining such sample data would be at most 10%, which is quite unlikely, so we reject the null hypothesis.
30 Lowertail test for the mean, variance unknown: example Example: 9.4 (cont.) in Excel: Go to menu: Data, submenu: Data Analysis, choose function: twosample ttest with unequal variances. Column A (data), Column B (n repetitions of µ 0 = 20), in yellow (observed t stat, pvalue and t n 1;α ).
31 Uppertail test for the variance: example Example: 9.5 (Newbold) In order to meet the standards in consignments of a chemical product, it is important that the variance of their percentage impurity levels does not exceed 4. A random sample of twenty consignments had a sample quasivariance of 5.62 for impurity level percentages. a) Perform a suitable test of hypothesis (α = 0.1). b) Find the power of the test. What is the power at σ 2 1 = 7? c) What sample size would guarantee a power of 0.9 at σ 2 1 = 7? Population: X = impurity level of a consignment of a chemical (in %s) X N(µ, σ 2 ) Test statistic: χ 2 = (n 1)s2 σ0 2 Observed test statistic: χ 2 n 1 'SRS: n = 20 Sample: s 2 = 5.62 Objective: test H 0 : σ 2 σ0 2 z} { 4 against H 1 : σ 2 > 4 (Uppertail test) σ 2 0 = 4 n = 20 s 2 = 5.62 χ 2 = (n 1)s2 σ0 2 = (20 1) =
32 Uppertail test for the variance: example Example: 9.5 a) (cont.) pvalue = P(χ ) (0.1, 0.25) because χ 2 19;0.25 {}}{ 22.7 < < χ 2 19;0.1 {}}{ 27.2 Hence, given that pvalue exceeds α = 0.1, we cannot reject the null hypothesis at this level. χ 2 n 1 density χ 2 = p value =area Conclusion: The sample data did not provide enough evidence to reject the claim that the variance of the percentage impurity levels in consignments of this chemical is at most 4.
33 Uppertail test for the variance: power Example: 9.5 b) Recall that: power = P(reject H 0 H 1 is true) When do we reject H 0? j ff (n 1)s 2 RR 0.1 = > χ 2 σ0 2 n 1;0.1 power(σ 2 ) versus σ = >< z } { >= = (n 1)s 2 > χ 2 n 1;0.1 σ0 2 >: >; Hence the power is: power(σ1) 2 = P reject H 0 σ 2 = σ1 2 = P (n 1)s 2 > σ 2 = σ1 2 (n 1)s 2 = P > «σ1 2 σ1 2 = P χ 2 > ««108.8 = 1 F χ 2 σ 2 1 σ α (F χ 2 is the cdf of χ 2 n 1) Hence, power(7) = P `χ 2 > σ 0 2 = 4 power(σ 2 = 1 β(σ 2 ) Θ 0 Θ 1 σ =
34 Uppertail test for the variance: sample size calculations Example: 9.5 c) From our previous calculations, ( we know that ) potencia(σ1 2) = P (n 1)s 2 > χ 2 σ 2 0 n 1;0.1, σ 2 1 Our objective is to find the smallest n such that: σ 2 1 (n 1)s 2 σ 2 1 χ 2 n {}}{ (n 1)s 2 power(7) = P > χ 2 4 n 1; σ 2 1 The last equation implies that we are dealing with a χ 2 n 1 distribution, whose upper 0.9quantile satisfies χ 2 n 1; χ2 n 1;0.1. chisquare χ table 2 43;0.9 /χ2 43;0.1 = > n 1 = 43 Thus, if we collect 44 observations we should be able to detect the alternative value σ1 2 = 7 with at least 90% chance.
35 Another power example: lowertail test for the mean, normal population, known σ α H 0 : µ µ 0 versus H 1 : µ < µ 0 at α = 0.05 Say that µ 0 = 5, n = 16, σ = 0.1 We reject H 0 if x µ0 σ/ n < z α = that is when x 4.96, hence ( ) power(µ 1 ) = P Z < 4.96 µ1 0.1/ 16 power(µ) = 1 β(µ) µ 0 = Θ 0 Θ 1 µ n=16 n=9 n=4
36 Another power example: lowertail test for the mean, normal population, known σ 2 Note that the power = 1 P(Type II error) function has the following features (everything else being equal): The farther the true mean µ 1 from the hypothesized µ 0, the greater the power The smaller the α, the smaller the power, that is, reducing the probability of Type I error will increase the probability of Type II error The larger the population variance, the lower the power (we are less likely to detect small departures from µ 0, when there is greater variability in the population) The larger the sample size, the greater the power of the test (the more info from the population, the greater the chance of detecting any departures from the null hypothesis).
HYPOTHESIS TESTING: POWER OF THE TEST
HYPOTHESIS TESTING: POWER OF THE TEST The first 6 steps of the 9step test of hypothesis are called "the test". These steps are not dependent on the observed data values. When planning a research project,
More information3.4 Statistical inference for 2 populations based on two samples
3.4 Statistical inference for 2 populations based on two samples Tests for a difference between two population means The first sample will be denoted as X 1, X 2,..., X m. The second sample will be denoted
More informationHypothesis testing  Steps
Hypothesis testing  Steps Steps to do a twotailed test of the hypothesis that β 1 0: 1. Set up the hypotheses: H 0 : β 1 = 0 H a : β 1 0. 2. Compute the test statistic: t = b 1 0 Std. error of b 1 =
More informationHypothesis Testing  One Mean
Hypothesis Testing  One Mean A hypothesis is simply a statement that something is true. Typically, there are two hypotheses in a hypothesis test: the null, and the alternative. Null Hypothesis The hypothesis
More informationChapter 7 Notes  Inference for Single Samples. You know already for a large sample, you can invoke the CLT so:
Chapter 7 Notes  Inference for Single Samples You know already for a large sample, you can invoke the CLT so: X N(µ, ). Also for a large sample, you can replace an unknown σ by s. You know how to do a
More informationChapter 4 Statistical Inference in Quality Control and Improvement. Statistical Quality Control (D. C. Montgomery)
Chapter 4 Statistical Inference in Quality Control and Improvement 許 湘 伶 Statistical Quality Control (D. C. Montgomery) Sampling distribution I a random sample of size n: if it is selected so that the
More informationIntroduction to Hypothesis Testing
I. Terms, Concepts. Introduction to Hypothesis Testing A. In general, we do not know the true value of population parameters  they must be estimated. However, we do have hypotheses about what the true
More informationLAB 4 INSTRUCTIONS CONFIDENCE INTERVALS AND HYPOTHESIS TESTING
LAB 4 INSTRUCTIONS CONFIDENCE INTERVALS AND HYPOTHESIS TESTING In this lab you will explore the concept of a confidence interval and hypothesis testing through a simulation problem in engineering setting.
More informationChapter 8 Hypothesis Testing Chapter 8 Hypothesis Testing 81 Overview 82 Basics of Hypothesis Testing
Chapter 8 Hypothesis Testing 1 Chapter 8 Hypothesis Testing 81 Overview 82 Basics of Hypothesis Testing 83 Testing a Claim About a Proportion 85 Testing a Claim About a Mean: s Not Known 86 Testing
More informationBA 275 Review Problems  Week 6 (10/30/0611/3/06) CD Lessons: 53, 54, 55, 56 Textbook: pp. 394398, 404408, 410420
BA 275 Review Problems  Week 6 (10/30/0611/3/06) CD Lessons: 53, 54, 55, 56 Textbook: pp. 394398, 404408, 410420 1. Which of the following will increase the value of the power in a statistical test
More informationEstimation of σ 2, the variance of ɛ
Estimation of σ 2, the variance of ɛ The variance of the errors σ 2 indicates how much observations deviate from the fitted surface. If σ 2 is small, parameters β 0, β 1,..., β k will be reliably estimated
More informationIntroduction to Hypothesis Testing OPRE 6301
Introduction to Hypothesis Testing OPRE 6301 Motivation... The purpose of hypothesis testing is to determine whether there is enough statistical evidence in favor of a certain belief, or hypothesis, about
More informationBusiness Statistics, 9e (Groebner/Shannon/Fry) Chapter 9 Introduction to Hypothesis Testing
Business Statistics, 9e (Groebner/Shannon/Fry) Chapter 9 Introduction to Hypothesis Testing 1) Hypothesis testing and confidence interval estimation are essentially two totally different statistical procedures
More informationTHE FIRST SET OF EXAMPLES USE SUMMARY DATA... EXAMPLE 7.2, PAGE 227 DESCRIBES A PROBLEM AND A HYPOTHESIS TEST IS PERFORMED IN EXAMPLE 7.
THERE ARE TWO WAYS TO DO HYPOTHESIS TESTING WITH STATCRUNCH: WITH SUMMARY DATA (AS IN EXAMPLE 7.17, PAGE 236, IN ROSNER); WITH THE ORIGINAL DATA (AS IN EXAMPLE 8.5, PAGE 301 IN ROSNER THAT USES DATA FROM
More informationHypothesis Testing. Hypothesis Testing
Hypothesis Testing Daniel A. Menascé Department of Computer Science George Mason University 1 Hypothesis Testing Purpose: make inferences about a population parameter by analyzing differences between observed
More information22. HYPOTHESIS TESTING
22. HYPOTHESIS TESTING Often, we need to make decisions based on incomplete information. Do the data support some belief ( hypothesis ) about the value of a population parameter? Is OJ Simpson guilty?
More informationPractice problems for Homework 12  confidence intervals and hypothesis testing. Open the Homework Assignment 12 and solve the problems.
Practice problems for Homework 1  confidence intervals and hypothesis testing. Read sections 10..3 and 10.3 of the text. Solve the practice problems below. Open the Homework Assignment 1 and solve the
More informationSummary of Formulas and Concepts. Descriptive Statistics (Ch. 14)
Summary of Formulas and Concepts Descriptive Statistics (Ch. 14) Definitions Population: The complete set of numerical information on a particular quantity in which an investigator is interested. We assume
More informationIntroduction. Hypothesis Testing. Hypothesis Testing. Significance Testing
Introduction Hypothesis Testing Mark Lunt Arthritis Research UK Centre for Ecellence in Epidemiology University of Manchester 13/10/2015 We saw last week that we can never know the population parameters
More informationClass 19: Two Way Tables, Conditional Distributions, ChiSquare (Text: Sections 2.5; 9.1)
Spring 204 Class 9: Two Way Tables, Conditional Distributions, ChiSquare (Text: Sections 2.5; 9.) Big Picture: More than Two Samples In Chapter 7: We looked at quantitative variables and compared the
More informationSection 7.1. Introduction to Hypothesis Testing. Schrodinger s cat quantum mechanics thought experiment (1935)
Section 7.1 Introduction to Hypothesis Testing Schrodinger s cat quantum mechanics thought experiment (1935) Statistical Hypotheses A statistical hypothesis is a claim about a population. Null hypothesis
More informationC. The null hypothesis is not rejected when the alternative hypothesis is true. A. population parameters.
Sample Multiple Choice Questions for the material since Midterm 2. Sample questions from Midterms and 2 are also representative of questions that may appear on the final exam.. A randomly selected sample
More informationConfidence Intervals for One Standard Deviation Using Standard Deviation
Chapter 640 Confidence Intervals for One Standard Deviation Using Standard Deviation Introduction This routine calculates the sample size necessary to achieve a specified interval width or distance from
More information1. What is the critical value for this 95% confidence interval? CV = z.025 = invnorm(0.025) = 1.96
1 Final Review 2 Review 2.1 CI 1propZint Scenario 1 A TV manufacturer claims in its warranty brochure that in the past not more than 10 percent of its TV sets needed any repair during the first two years
More informationConfidence Intervals for Cp
Chapter 296 Confidence Intervals for Cp Introduction This routine calculates the sample size needed to obtain a specified width of a Cp confidence interval at a stated confidence level. Cp is a process
More informationHYPOTHESIS TESTING (ONE SAMPLE)  CHAPTER 7 1. used confidence intervals to answer questions such as...
HYPOTHESIS TESTING (ONE SAMPLE)  CHAPTER 7 1 PREVIOUSLY used confidence intervals to answer questions such as... You know that 0.25% of women have red/green color blindness. You conduct a study of men
More informationUnit 26 Estimation with Confidence Intervals
Unit 26 Estimation with Confidence Intervals Objectives: To see how confidence intervals are used to estimate a population proportion, a population mean, a difference in population proportions, or a difference
More informationPermutation Tests for Comparing Two Populations
Permutation Tests for Comparing Two Populations Ferry Butar Butar, Ph.D. JaeWan Park Abstract Permutation tests for comparing two populations could be widely used in practice because of flexibility of
More informationHypothesis testing. c 2014, Jeffrey S. Simonoff 1
Hypothesis testing So far, we ve talked about inference from the point of estimation. We ve tried to answer questions like What is a good estimate for a typical value? or How much variability is there
More informationComparing Means in Two Populations
Comparing Means in Two Populations Overview The previous section discussed hypothesis testing when sampling from a single population (either a single mean or two means from the same population). Now we
More informationLesson 1: Comparison of Population Means Part c: Comparison of Two Means
Lesson : Comparison of Population Means Part c: Comparison of Two Means Welcome to lesson c. This third lesson of lesson will discuss hypothesis testing for two independent means. Steps in Hypothesis
More informationTwoSample TTests Assuming Equal Variance (Enter Means)
Chapter 4 TwoSample TTests Assuming Equal Variance (Enter Means) Introduction This procedure provides sample size and power calculations for one or twosided twosample ttests when the variances of
More informationDifference of Means and ANOVA Problems
Difference of Means and Problems Dr. Tom Ilvento FREC 408 Accounting Firm Study An accounting firm specializes in auditing the financial records of large firm It is interested in evaluating its fee structure,particularly
More informationIntroduction to. Hypothesis Testing CHAPTER LEARNING OBJECTIVES. 1 Identify the four steps of hypothesis testing.
Introduction to Hypothesis Testing CHAPTER 8 LEARNING OBJECTIVES After reading this chapter, you should be able to: 1 Identify the four steps of hypothesis testing. 2 Define null hypothesis, alternative
More informationExperimental Design. Power and Sample Size Determination. Proportions. Proportions. Confidence Interval for p. The Binomial Test
Experimental Design Power and Sample Size Determination Bret Hanlon and Bret Larget Department of Statistics University of Wisconsin Madison November 3 8, 2011 To this point in the semester, we have largely
More informationMath 251, Review Questions for Test 3 Rough Answers
Math 251, Review Questions for Test 3 Rough Answers 1. (Review of some terminology from Section 7.1) In a state with 459,341 voters, a poll of 2300 voters finds that 45 percent support the Republican candidate,
More informationHYPOTHESIS TESTING (ONE SAMPLE)  CHAPTER 7 1. used confidence intervals to answer questions such as...
HYPOTHESIS TESTING (ONE SAMPLE)  CHAPTER 7 1 PREVIOUSLY used confidence intervals to answer questions such as... You know that 0.25% of women have red/green color blindness. You conduct a study of men
More informationIntroduction to Hypothesis Testing. Hypothesis Testing. Step 1: State the Hypotheses
Introduction to Hypothesis Testing 1 Hypothesis Testing A hypothesis test is a statistical procedure that uses sample data to evaluate a hypothesis about a population Hypothesis is stated in terms of the
More informationHow To Test For Significance On A Data Set
NonParametric Univariate Tests: 1 Sample Sign Test 1 1 SAMPLE SIGN TEST A nonparametric equivalent of the 1 SAMPLE TTEST. ASSUMPTIONS: Data is nonnormally distributed, even after log transforming.
More informationNormal distribution. ) 2 /2σ. 2π σ
Normal distribution The normal distribution is the most widely known and used of all distributions. Because the normal distribution approximates many natural phenomena so well, it has developed into a
More informationTests for Two Proportions
Chapter 200 Tests for Two Proportions Introduction This module computes power and sample size for hypothesis tests of the difference, ratio, or odds ratio of two independent proportions. The test statistics
More informationTwoSample TTests Allowing Unequal Variance (Enter Difference)
Chapter 45 TwoSample TTests Allowing Unequal Variance (Enter Difference) Introduction This procedure provides sample size and power calculations for one or twosided twosample ttests when no assumption
More informationComparing Two Groups. Standard Error of ȳ 1 ȳ 2. Setting. Two Independent Samples
Comparing Two Groups Chapter 7 describes two ways to compare two populations on the basis of independent samples: a confidence interval for the difference in population means and a hypothesis test. The
More information12.5: CHISQUARE GOODNESS OF FIT TESTS
125: ChiSquare Goodness of Fit Tests CD121 125: CHISQUARE GOODNESS OF FIT TESTS In this section, the χ 2 distribution is used for testing the goodness of fit of a set of data to a specific probability
More informationLecture Notes Module 1
Lecture Notes Module 1 Study Populations A study population is a clearly defined collection of people, animals, plants, or objects. In psychological research, a study population usually consists of a specific
More information4. Continuous Random Variables, the Pareto and Normal Distributions
4. Continuous Random Variables, the Pareto and Normal Distributions A continuous random variable X can take any value in a given range (e.g. height, weight, age). The distribution of a continuous random
More informationRecall this chart that showed how most of our course would be organized:
Chapter 4 OneWay ANOVA Recall this chart that showed how most of our course would be organized: Explanatory Variable(s) Response Variable Methods Categorical Categorical Contingency Tables Categorical
More informationConfidence Intervals for the Difference Between Two Means
Chapter 47 Confidence Intervals for the Difference Between Two Means Introduction This procedure calculates the sample size necessary to achieve a specified distance from the difference in sample means
More informationAn Introduction to Statistics Course (ECOE 1302) Spring Semester 2011 Chapter 10 TWOSAMPLE TESTS
The Islamic University of Gaza Faculty of Commerce Department of Economics and Political Sciences An Introduction to Statistics Course (ECOE 130) Spring Semester 011 Chapter 10 TWOSAMPLE TESTS Practice
More informationp ˆ (sample mean and sample
Chapter 6: Confidence Intervals and Hypothesis Testing When analyzing data, we can t just accept the sample mean or sample proportion as the official mean or proportion. When we estimate the statistics
More information1 Sufficient statistics
1 Sufficient statistics A statistic is a function T = rx 1, X 2,, X n of the random sample X 1, X 2,, X n. Examples are X n = 1 n s 2 = = X i, 1 n 1 the sample mean X i X n 2, the sample variance T 1 =
More informationSimple Linear Regression Inference
Simple Linear Regression Inference 1 Inference requirements The Normality assumption of the stochastic term e is needed for inference even if it is not a OLS requirement. Therefore we have: Interpretation
More informationCHAPTER 14 NONPARAMETRIC TESTS
CHAPTER 14 NONPARAMETRIC TESTS Everything that we have done up until now in statistics has relied heavily on one major fact: that our data is normally distributed. We have been able to make inferences
More informationBA 275 Review Problems  Week 5 (10/23/0610/27/06) CD Lessons: 48, 49, 50, 51, 52 Textbook: pp. 380394
BA 275 Review Problems  Week 5 (10/23/0610/27/06) CD Lessons: 48, 49, 50, 51, 52 Textbook: pp. 380394 1. Does vigorous exercise affect concentration? In general, the time needed for people to complete
More informationHypothesis Testing for Beginners
Hypothesis Testing for Beginners Michele Piffer LSE August, 2011 Michele Piffer (LSE) Hypothesis Testing for Beginners August, 2011 1 / 53 One year ago a friend asked me to put down some easytoread notes
More informationLecture 8. Confidence intervals and the central limit theorem
Lecture 8. Confidence intervals and the central limit theorem Mathematical Statistics and Discrete Mathematics November 25th, 2015 1 / 15 Central limit theorem Let X 1, X 2,... X n be a random sample of
More informationChapter 4: Statistical Hypothesis Testing
Chapter 4: Statistical Hypothesis Testing Christophe Hurlin November 20, 2015 Christophe Hurlin () Advanced Econometrics  Master ESA November 20, 2015 1 / 225 Section 1 Introduction Christophe Hurlin
More informationSTART Selected Topics in Assurance
START Selected Topics in Assurance Related Technologies Table of Contents Introduction Some Statistical Background Fitting a Normal Using the Anderson Darling GoF Test Fitting a Weibull Using the Anderson
More informationCalculating PValues. Parkland College. Isela Guerra Parkland College. Recommended Citation
Parkland College A with Honors Projects Honors Program 2014 Calculating PValues Isela Guerra Parkland College Recommended Citation Guerra, Isela, "Calculating PValues" (2014). A with Honors Projects.
More informationNonInferiority Tests for Two Proportions
Chapter 0 NonInferiority Tests for Two Proportions Introduction This module provides power analysis and sample size calculation for noninferiority and superiority tests in twosample designs in which
More informationHow To Check For Differences In The One Way Anova
MINITAB ASSISTANT WHITE PAPER This paper explains the research conducted by Minitab statisticians to develop the methods and data checks used in the Assistant in Minitab 17 Statistical Software. OneWay
More informationNonInferiority Tests for Two Means using Differences
Chapter 450 oninferiority Tests for Two Means using Differences Introduction This procedure computes power and sample size for noninferiority tests in twosample designs in which the outcome is a continuous
More information5/31/2013. Chapter 8 Hypothesis Testing. Hypothesis Testing. Hypothesis Testing. Outline. Objectives. Objectives
C H 8A P T E R Outline 8 1 Steps in Traditional Method 8 2 z Test for a Mean 8 3 t Test for a Mean 8 4 z Test for a Proportion 8 6 Confidence Intervals and Copyright 2013 The McGraw Hill Companies, Inc.
More informationChapter 8: Hypothesis Testing for One Population Mean, Variance, and Proportion
Chapter 8: Hypothesis Testing for One Population Mean, Variance, and Proportion Learning Objectives Upon successful completion of Chapter 8, you will be able to: Understand terms. State the null and alternative
More informationStats Review Chapters 910
Stats Review Chapters 910 Created by Teri Johnson Math Coordinator, Mary Stangler Center for Academic Success Examples are taken from Statistics 4 E by Michael Sullivan, III And the corresponding Test
More informationInference for two Population Means
Inference for two Population Means Bret Hanlon and Bret Larget Department of Statistics University of Wisconsin Madison October 27 November 1, 2011 Two Population Means 1 / 65 Case Study Case Study Example
More informationTests for One Proportion
Chapter 100 Tests for One Proportion Introduction The OneSample Proportion Test is used to assess whether a population proportion (P1) is significantly different from a hypothesized value (P0). This is
More information1.5 Oneway Analysis of Variance
Statistics: Rosie Cornish. 200. 1.5 Oneway Analysis of Variance 1 Introduction Oneway analysis of variance (ANOVA) is used to compare several means. This method is often used in scientific or medical experiments
More informationPoint and Interval Estimates
Point and Interval Estimates Suppose we want to estimate a parameter, such as p or µ, based on a finite sample of data. There are two main methods: 1. Point estimate: Summarize the sample by a single number
More informationName: Date: Use the following to answer questions 34:
Name: Date: 1. Determine whether each of the following statements is true or false. A) The margin of error for a 95% confidence interval for the mean increases as the sample size increases. B) The margin
More informationOdds ratio, Odds ratio test for independence, chisquared statistic.
Odds ratio, Odds ratio test for independence, chisquared statistic. Announcements: Assignment 5 is live on webpage. Due Wed Aug 1 at 4:30pm. (9 days, 1 hour, 58.5 minutes ) Final exam is Aug 9. Review
More informationPsychology 60 Fall 2013 Practice Exam Actual Exam: Next Monday. Good luck!
Psychology 60 Fall 2013 Practice Exam Actual Exam: Next Monday. Good luck! Name: 1. The basic idea behind hypothesis testing: A. is important only if you want to compare two populations. B. depends on
More informationTopic 8. Chi Square Tests
BE540W Chi Square Tests Page 1 of 5 Topic 8 Chi Square Tests Topics 1. Introduction to Contingency Tables. Introduction to the Contingency Table Hypothesis Test of No Association.. 3. The Chi Square Test
More informationTesting Hypotheses About Proportions
Chapter 11 Testing Hypotheses About Proportions Hypothesis testing method: uses data from a sample to judge whether or not a statement about a population may be true. Steps in Any Hypothesis Test 1. Determine
More information5.1 Identifying the Target Parameter
University of California, Davis Department of Statistics Summer Session II Statistics 13 August 20, 2012 Date of latest update: August 20 Lecture 5: Estimation with Confidence intervals 5.1 Identifying
More informationReview #2. Statistics
Review #2 Statistics Find the mean of the given probability distribution. 1) x P(x) 0 0.19 1 0.37 2 0.16 3 0.26 4 0.02 A) 1.64 B) 1.45 C) 1.55 D) 1.74 2) The number of golf balls ordered by customers of
More informationTutorial 5: Hypothesis Testing
Tutorial 5: Hypothesis Testing Rob Nicholls nicholls@mrclmb.cam.ac.uk MRC LMB Statistics Course 2014 Contents 1 Introduction................................ 1 2 Testing distributional assumptions....................
More informationFairfield Public Schools
Mathematics Fairfield Public Schools AP Statistics AP Statistics BOE Approved 04/08/2014 1 AP STATISTICS Critical Areas of Focus AP Statistics is a rigorous course that offers advanced students an opportunity
More informationStat 411/511 THE RANDOMIZATION TEST. Charlotte Wickham. stat511.cwick.co.nz. Oct 16 2015
Stat 411/511 THE RANDOMIZATION TEST Oct 16 2015 Charlotte Wickham stat511.cwick.co.nz Today Review randomization model Conduct randomization test What about CIs? Using a tdistribution as an approximation
More informationNCSS Statistical Software
Chapter 06 Introduction This procedure provides several reports for the comparison of two distributions, including confidence intervals for the difference in means, twosample ttests, the ztest, the
More informationHypothesis Testing. Steps for a hypothesis test:
Hypothesis Testing Steps for a hypothesis test: 1. State the claim H 0 and the alternative, H a 2. Choose a significance level or use the given one. 3. Draw the sampling distribution based on the assumption
More informationNovember 08, 2010. 155S8.6_3 Testing a Claim About a Standard Deviation or Variance
Chapter 8 Hypothesis Testing 8 1 Review and Preview 8 2 Basics of Hypothesis Testing 8 3 Testing a Claim about a Proportion 8 4 Testing a Claim About a Mean: σ Known 8 5 Testing a Claim About a Mean: σ
More informationSolutions to Questions on Hypothesis Testing and Regression
Solutions to Questions on Hypothesis Testing and Regression 1. A mileage test is conducted for a new car model, the Pizzazz. Thirty (n=30) random selected Pizzazzes are driven for a month and the mileage
More informationTwo Correlated Proportions (McNemar Test)
Chapter 50 Two Correlated Proportions (Mcemar Test) Introduction This procedure computes confidence intervals and hypothesis tests for the comparison of the marginal frequencies of two factors (each with
More informationUnit 31 A Hypothesis Test about Correlation and Slope in a Simple Linear Regression
Unit 31 A Hypothesis Test about Correlation and Slope in a Simple Linear Regression Objectives: To perform a hypothesis test concerning the slope of a least squares line To recognize that testing for a
More informationOutline. Topic 4  Analysis of Variance Approach to Regression. Partitioning Sums of Squares. Total Sum of Squares. Partitioning sums of squares
Topic 4  Analysis of Variance Approach to Regression Outline Partitioning sums of squares Degrees of freedom Expected mean squares General linear test  Fall 2013 R 2 and the coefficient of correlation
More informationCHISQUARE: TESTING FOR GOODNESS OF FIT
CHISQUARE: TESTING FOR GOODNESS OF FIT In the previous chapter we discussed procedures for fitting a hypothesized function to a set of experimental data points. Such procedures involve minimizing a quantity
More informationConfidence Intervals for Cpk
Chapter 297 Confidence Intervals for Cpk Introduction This routine calculates the sample size needed to obtain a specified width of a Cpk confidence interval at a stated confidence level. Cpk is a process
More informationThe Wilcoxon RankSum Test
1 The Wilcoxon RankSum Test The Wilcoxon ranksum test is a nonparametric alternative to the twosample ttest which is based solely on the order in which the observations from the two samples fall. We
More informationSTAT 350 Practice Final Exam Solution (Spring 2015)
PART 1: Multiple Choice Questions: 1) A study was conducted to compare five different training programs for improving endurance. Forty subjects were randomly divided into five groups of eight subjects
More informationt Tests in Excel The Excel Statistical Master By Mark Harmon Copyright 2011 Mark Harmon
ttests in Excel By Mark Harmon Copyright 2011 Mark Harmon No part of this publication may be reproduced or distributed without the express permission of the author. mark@excelmasterseries.com www.excelmasterseries.com
More informationName: (b) Find the minimum sample size you should use in order for your estimate to be within 0.03 of p when the confidence level is 95%.
Chapter 78 Exam Name: Answer the questions in the spaces provided. If you run out of room, show your work on a separate paper clearly numbered and attached to this exam. Please indicate which program
More informationExact Confidence Intervals
Math 541: Statistical Theory II Instructor: Songfeng Zheng Exact Confidence Intervals Confidence intervals provide an alternative to using an estimator ˆθ when we wish to estimate an unknown parameter
More informationHypothesis Testing. Reminder of Inferential Statistics. Hypothesis Testing: Introduction
Hypothesis Testing PSY 360 Introduction to Statistics for the Behavioral Sciences Reminder of Inferential Statistics All inferential statistics have the following in common: Use of some descriptive statistic
More informationUsing Stata for One Sample Tests
Using Stata for One Sample Tests All of the one sample problems we have discussed so far can be solved in Stata via either (a) statistical calculator functions, where you provide Stata with the necessary
More informationTwosample inference: Continuous data
Twosample inference: Continuous data Patrick Breheny April 5 Patrick Breheny STA 580: Biostatistics I 1/32 Introduction Our next two lectures will deal with twosample inference for continuous data As
More informationSAMPLE SIZE CONSIDERATIONS
SAMPLE SIZE CONSIDERATIONS Learning Objectives Understand the critical role having the right sample size has on an analysis or study. Know how to determine the correct sample size for a specific study.
More informationExact Nonparametric Tests for Comparing Means  A Personal Summary
Exact Nonparametric Tests for Comparing Means  A Personal Summary Karl H. Schlag European University Institute 1 December 14, 2006 1 Economics Department, European University Institute. Via della Piazzuola
More informationPractice Problems and Exams
Practice Problems and Exams 1 The Islamic University of Gaza Faculty of Commerce Department of Economics and Political Sciences An Introduction to Statistics Course (ECOE 1302) Spring Semester 20092010
More information